Surface Gravity and Normalization

Surface gravity is the non-affinity of a chosen Killing generator on a Killing horizon. Several formulas compute it, but they agree only after the horizon and the normalization of its generator have been fixed. This page derives the equivalence, shows how $\kappa$ rescales, and explains why a quoted value without a time normalization is incomplete.

Required background. Killing Fields and Isometry Groups supplies the Killing equation and normalization of symmetry generators.

Helpful background. Exact Solutions of Einstein’s Equations supplies regular horizon charts; Causality and Global Structure supplies the distinction between Killing and event horizons; The Einstein Equation is needed only for later mechanics laws; Black-Hole Geometry and Mechanics provides the horizon taxonomy.

Non-affinity and the norm-gradient definition

Let $\mathcal H$ be a smooth Killing horizon generated by a Killing field $\chi^a$. Choose $\chi^a$ future-directed on the future horizon. Its integral curves are null geodesics, generally with a non-affine Killing parameter $v$:

$$\chi^b\nabla_b\chi^a=\kappa\chi^a \quad\text{on }\mathcal H.$$

This equation defines $\kappa$. Lowering the free index and using $\nabla_{(a}\chi_{b)}=0$ gives

$$\begin{aligned} \chi^b\nabla_b\chi_a &=-\chi^b\nabla_a\chi_b \\ &=-\frac12\nabla_a(\chi^2), \end{aligned}$$

and hence the equivalent norm-gradient formula

$$\nabla_a(\chi^2)=-2\kappa\chi_a \quad\text{on }\mathcal H.$$

The proportionality is possible because both sides are normal covectors to the horizon. It is also a useful diagnostic: if the gradient of $\chi^2$ is not parallel to $\chi_a$ on a null level set, the proposed field is not its normal Killing generator. Rácz and Wald state these definitions and their geometric identities in Rácz and Wald 1996, arXiv v1, §2, pp. 5–7.

For a regular Killing horizon, another invariant formula is

$$\kappa^2 =-\frac12(\nabla_a\chi_b)(\nabla^a\chi^b) \quad\text{on }\mathcal H.$$

At a bifurcation surface this follows transparently: $\chi^a=0$, while $\nabla_a\chi_b$ generates a Lorentz boost in the two-dimensional normal plane and equals $\kappa$ times its binormal. Killing-flow invariance carries the scalar along each Killing orbit. It does not by itself make $\kappa$ equal on different generators; that is the separate zeroth-law question. The squared formula determines $|\kappa|$; time orientation and the chosen outward direction determine its sign.

Static lapse and near-horizon slope

In a static region, write $\chi^2=-N^2$, with $N\gt0$ outside the horizon. The four-velocity of a stationary observer is $u^a=\chi^a/N$. Its acceleration satisfies

$$a_a=u^b\nabla_bu_a=\nabla_a\ln N.$$

Although $|a|$ diverges as $N\to0$, the redshifted acceleration has a finite limit on a nondegenerate horizon:

$$\kappa=\lim_{\mathcal H}N\sqrt{a^a a_a},$$

with the positive sign for the standard future-directed outer-horizon normalization. This formula has a direct operational interpretation in a static exterior, but it is not the definition for a general rotating horizon. There, $\chi^a=t^a+\Omega_H\phi^a$ rather than $t^a$ alone.

For a regular ingoing chart

$$ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2d\Omega_2^2,$$

with a simple horizon root $f(r_H)=0$, the generator is $\chi^a=(\partial_v)^a$. Since $\chi^2=-f$, the norm-gradient equation gives

$$\kappa=\frac12f'(r_H).$$

For Schwarzschild, $f=1-2GM/r$ and $r_H=2GM$, so

$$\kappa=\frac{1}{4GM}.$$

This has dimension inverse length, as required in units with $c=1$. The regular-coordinate calculation and the same value appear in Gourgoulhon and Jaramillo 2006, arXiv v2, Example 2.5, pp. 24–28.

What a normalization changes

Let $\chi'^a=c\chi^a$ for a nonzero constant $c$. The new field is still Killing, but

$$\kappa'=c\kappa.$$

The unparametrized null generators and the hypersurface do not change. Only the rate at which the group parameter advances along each orbit changes. A nonconstant rescaling would be allowed for a generic null normal, with $\kappa'=f(\kappa+\chi\cdot\nabla\ln f)$, but it generally ceases to be a spacetime Killing field.

In an asymptotically flat stationary black hole, one normally chooses $t^a$ to approach unit time translation at the selected infinity and chooses $\phi^a$ to have $2\pi$-periodic orbits. Writing $\chi^a=t^a+\Omega_H\phi^a$ then fixes the scale of $\chi^a$ and therefore of $\kappa$. In anti-de Sitter applications, a boundary metric and a nonrotating boundary frame play the analogous role. A local Rindler horizon has no preferred asymptotic normalization, so its numerical surface gravity depends on the boost parameter chosen by the observer family.

The classical black-hole mechanics formulas use a generator normalized in this way; their original stationary axisymmetric form is developed by Bardeen, Carter, and Hawking 1973, pp. 161–170. No quantum thermal interpretation is needed for the geometric statements here.

Sign, constancy, and degeneracy

Three adjectives answer different questions. Normalized says how the Killing parameter is calibrated. Constant says whether the resulting $\kappa$ has the same value on all generators and cross-sections. Degenerate means $\kappa=0$. The Killing equation alone defines $\kappa$ but does not, in full generality, prove constancy across a connected horizon. Constancy follows under additional geometric or field-equation hypotheses, treated with the zeroth law later in this volume.

Changing $\chi^a$ to $-\chi^a$ reverses its time orientation and changes the sign of $\kappa$. For a future outer Schwarzschild horizon one conventionally takes $\chi^a$ future-directed and $\kappa\gt0$. The invariant squared formula cannot encode this orientation choice.

Rescaling diagnostic

The Rindler metric is

$$ds^2=-\alpha^2\rho^2dt^2+d\rho^2+dy^2+dz^2,$$

with generator $\chi^a=(\partial_t)^a$ and horizon $\rho=0$. Find its surface gravity, then replace $t$ by $t'=bt$ with $b\gt0$ and recompute using the new generator $(\partial_{t'})^a$.

Solution

The lapse is $N=\alpha\rho$, while a stationary observer has acceleration magnitude $1/\rho$. Therefore $N|a|=\alpha$ and $\kappa=\alpha$ for $\partial_t$. Since $\partial_{t'}=b^{-1}\partial_t$, the rescaled generator has $\kappa'=\alpha/b$. The spacetime and the horizon have not changed. The calculation isolates why a normalization prescription, not merely a null surface, is part of every numerical value of surface gravity.

Bifurcation Surfaces and Extensions next explains why a nonzero, constant surface gravity permits a regular bifurcate extension only under stated local or global hypotheses.

References

• Bardeen, James M., Brandon Carter, and Stephen W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. DOI.
• Gourgoulhon, Éric, and José Luis Jaramillo. “A 3+1 Perspective on Null Hypersurfaces and Isolated Horizons.” Physics Reports 423 (2006): 159–294. DOI. Open PDF.
• Rácz, István, and Robert M. Wald. “Global Extensions of Spacetimes Describing Asymptotic Final States of Black Holes.” Classical and Quantum Gravity 13 (1996): 539–553. DOI. Open PDF.